If `A={lambda, (lambda, mu)}`, then the power set of A is

Options

A. `{phi, {phi}, {lambda}, {lambda, mu}}`
B. `{phi,{lambda},{{lambda,mu}},{lambda,{lambda,mu}}}`
C. `{phi,{lambda},{lambda,mu},{lambda,{lambda,mu}}}`
D. `{{lambda},{lambda,mu},{lambda,{lambda,mu}}}`

Explanation

To find the power set of the set \( A = \{ \lambda, (\lambda, \mu) \} \), we will follow these steps:

### Step 1: Understand the Definition of Power Set
The power set of a set \( S \) is the set of all possible subsets of \( S \), including the empty set and \( S \) itself.

### Step 2: Identify the Elements of Set A
The set \( A \) has two elements:
1. \( \lambda \)
2. \( (\lambda, \mu) \)

### Step 3: List All Possible Subsets
Now, we will list all possible subsets of \( A \):
1. The empty set: \( \emptyset \)
2. The subset containing only the first element: \( \{ \lambda \} \)
3. The subset containing only the second element: \( \{ (\lambda, \mu) \} \)
4. The subset containing both elements: \( \{ \lambda, (\lambda, \mu) \} \)

### Step 4: Compile the Power Set
Now we can compile the power set \( P(A) \):
\[ P(A) = \{ \emptyset, \{ \lambda \}, \{ (\lambda, \mu) \}, \{ \lambda, (\lambda, \mu) \} \} \]

### Final Answer
Thus, the power set of \( A \) is:
\[ P(A) = \{ \emptyset, \{ \lambda \}, \{ (\lambda, \mu) \}, \{ \lambda, (\lambda, \mu) \} \} \]

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